Frobenius Reciprocity; Real and Quaternionic Representations — Studies in Algebra and Group Theory

37 Frobenius Reciprocity; Real and Quaternionic Representations

3 December 2025 · Denis Auroux

Last time

Induced representations. Let be a subgroup, and (invariant under ) (a representation of ). Denote the cosets of by . Then iff .

Concretely, each permutes the summands according to its action on : if , then maps to (and ).

Frobenius reciprocity

Proposition 37.1 (Frobenius reciprocity). If is a representation of , and a representation of , then every -equivariant map extends uniquely to a -equivariant map :

Proof. Choose representatives of the cosets of , and let . Given -equivariant, if is -equivariant and , then necessarily where is the action of on . This determines for all , hence , uniquely.

To check that is -equivariant, recall that acts on by mapping to such that , via . Given , where in , is acting on . So on for all , hence on .

So does have a unique -equivariant extension .

Conversely, given , is -equivariant, and hence its restriction to is -equivariant. ∎

Comparing dimensions, , which gives:

Corollary 37.2 (Frobenius reciprocity).

(This can also be proved by direct calculation using the formula for from last time.)

Thus: if is an irreducible representation of and an irreducible representation of , then the number of times appears in is equal to the number of times appears in .

Some of the key motivation for studying induced representations comes from two deep theorems of Artin and Brauer.

Theorem 37.3 (Artin). Every character of a representation of is a linear combination with rational coefficients of characters of representations induced from cyclic subgroups of .

Theorem 37.4 (Brauer). Every character of a representation of is a linear combination with integer coefficients of characters of representations induced from 1-dimensional representations of ”elementary” subgroups of ,

where elementary means isomorphic to a product , with a -group () and cyclic, , .

(We won’t prove these. See e.g. Serre’s Linear Representations of Finite Groups.)

Real representations

We’ve studied actions of finite groups on complex vector spaces; now we want to do the same for real ones.

Existence of an invariant inner product still holds (build by averaging). So every representation is a of irreducibles (given a subrepresentation, its is also a subrepresentation).
Schur’s lemma fails: (for ) acts on by rotations; this is irreducible, and has nontrivial automorphisms, e.g. any rotation of .

The main tool to study real representations is complexification: ( acts by ).

Definition 37.5. A complex representation of is called real if there exists a representation over , , such that .

A necessary condition: must take real values! (The trace of a matrix with entries is real!) This is alas not a sufficient condition.

Example 37.6. The quaternion group , , acts faithfully on by , and all the others have , so takes real values.

However, this does not come from a 2-dimensional real representation: . (This is because a real representation of a finite group has an invariant inner product, by the same averaging trick as in the complex case, so we’d get (since the action is faithful), with acting by ; but only 2 elements of square to (the rotations by ), while we need 6 such elements, for .)

Proposition 37.7. A complex representation is real iff there exists a -equivariant, complex antilinear map (i.e. ) such that .

Proof.

is clear: if , let for : complex conjugation!
: given , decomposes into which belong to the eigenspaces of . Let , which is an -subspace of (not a -subspace!), and, as -linear maps, , so is the eigenspace, and . The above was just linear algebra, but -equivariance of implies that the eigenspace is preserved by , hence is a subrepresentation over (similarly for ).

∎

If a complex representation is real, then acts by real matrices, so takes values in .

Conversely, let be an irreducible complex representation of such that takes values in .

Remark (Warning). Even if is irreducible over , might not be irreducible over : see e.g. acting on by rotations.

Then , so as -representations.

Recall: a linear map determines a bilinear form , . is -invariant iff is -equivariant. Thus Schur’s lemma for irreducible implies that admits a -invariant bilinear form , unique up to scaling, and nondegenerate if nonzero.

Now recall , i.e. the symmetric and skew parts of () are also -invariant bilinear forms on . By uniqueness, one of these is zero and the other is nondegenerate; i.e. is either symmetric or skew.

The symmetric case corresponds to real representations; the skew-symmetric case gives ”quaternionic” representations.

Proposition 37.8. An irreducible complex representation of a finite group is real iff carries a -invariant nondegenerate symmetric bilinear form .

Proof.

Assume is real. Then has an invariant real inner product ; extend it -bilinearly: defines a nondegenerate symmetric bilinear form on .
Conversely: determines an isomorphism (-linear, equivariant); choosing an invariant Hermitian inner product on , we also have a -antilinear equivariant bijection . Composing one with the inverse of the other gives a -antilinear equivariant map , characterized by is now an equivariant -linear isomorphism , hence by Schur. A calculation: shows ; replacing by , we can arrange . Thus is real by the previous proposition.

∎

Remark (Quaternionic representations). In the other case, where the invariant bilinear form is skew-symmetric, the same argument gives a -antilinear equivariant bijective map which now satisfies . This is a quaternionic structure on , i.e. it describes a structure of -module on , where a “division algebra” (a noncommutative analogue of a field: is a noncommutative ring such that every nonzero element has a multiplicative inverse). , with , , so an -module is the same thing as a -vector space plus an antilinear map such that .

Example 37.9. The standard representation of is real. This can be seen directly if we notice that acts on by rotations and reflections, and … or more abstractly by observing that , and has no trivial summand, hence there is no invariant skew-symmetric , but has a trivial summand, giving an invariant symmetric bilinear form , and applying the above.

Example 37.10. The quaternion group , , acts on by takes real values, but this doesn’t come from a real representation: .

Rather, this is a quaternionic representation: , and the above linear maps correspond to left multiplication by elements of . (E.g. , .) The -antilinear map , , is right multiplication by (it commutes with left multiplication ✓).

We’ll end here with the content on representation theory.